scholarly journals The estimation of approximation error using inverse problem and a set of numerical solutions

2021 ◽  
pp. 1-17
Author(s):  
A. K. Alekseev ◽  
A. E. Bondarev
Keyword(s):  
Inverse Problem ◽  
Numerical Solutions ◽  
Approximation Error

A Meshless Method for Solving the Mathematical Model Associated the Leakage Problem in Gas Pipeline

2014 ◽  
Vol 986-987 ◽  
pp. 1418-1421
Author(s):  
Jun Shan Li
Keyword(s):  
Differential Equation ◽  
Mathematical Model ◽  
Partial Differential Equation ◽  
Inverse Problem ◽  
Basis Function ◽  
Meshless Method ◽  
Numerical Solutions ◽  
Basis Functions ◽  
Gas Pipeline ◽  
The Mathematical Model

In this paper, we propose a meshless method for solving the mathematical model concerning the leakage problem when the pressure is tested in the gas pipeline. The method of radial basis function (RBF) can be used for solving partial differential equation by writing the solution in the form of linear combination of radius basis functions, that is, when integrating the definite conditions, one can find the combination coefficients and then the numerical solution. The leak problem is a kind of inverse problem that is focused by many engineers or mathematical researchers. The strength of the leak can find easily by the additional conditions and the numerical solutions.

scholarly journals Stochastic modeling error reduction using Bayesian approach coupled with an adaptive Kriging-based model

2014 ◽  
Vol 33 (3) ◽  
pp. 856-867 ◽  
Author(s):  
Ahmed Abou-Elyazied Abdallh ◽  
Luc Dupré
Keyword(s):  
Inverse Problem ◽  
Stochastic Modeling ◽  
Approximation Error ◽  
Memory Storage ◽  
Computational Time ◽  
Problem Solution ◽  
Content Type ◽  
Inverse Problem Solution ◽  
Bayesian Approximation ◽  
Adaptive Kriging

Purpose – Magnetic material properties of an electromagnetic device (EMD) can be recovered by solving a coupled experimental numerical inverse problem. In order to ensure the highest possible accuracy of the inverse problem solution, all physics of the EMD need to be perfectly modeled using a complex numerical model. However, these fine models demand a high computational time. Alternatively, less accurate coarse models can be used with a demerit of the high expected recovery errors. The purpose of this paper is to present an efficient methodology to reduce the effect of stochastic modeling errors in the inverse problem solution. Design/methodology/approach – The recovery error in the electromagnetic inverse problem solution is reduced using the Bayesian approximation error approach coupled with an adaptive Kriging-based model. The accuracy of the forward model is assessed and adapted a priori using the cross-validation technique. Findings – The adaptive Kriging-based model seems to be an efficient technique for modeling EMDs used in inverse problems. Moreover, using the proposed methodology, the recovery error in the electromagnetic inverse problem solution is largely reduced in a relatively small computational time and memory storage. Originality/value – The proposed methodology is capable of not only improving the accuracy of the inverse problem solution, but also reducing the computational time as well as the memory storage. Furthermore, to the best of the authors knowledge, it is the first time to combine the adaptive Kriging-based model with the Bayesian approximation error approach for the stochastic modeling error reduction.

Identification of nonlinear B–H curves based on magnetic field computations and multigrid methods for ill-posed problems

2003 ◽  
Vol 14 (1) ◽  
pp. 15-38 ◽  
Author(s):  
BARBARA KALTENBACHER ◽  
MANFRED KALTENBACHER ◽  
STEFAN REITZINGER
Keyword(s):  
Magnetic Field ◽  
Inverse Problem ◽  
Numerical Solutions ◽  
Iterative Scheme ◽  
Three Dimensional ◽  
Multigrid Methods ◽  
Stopping Criterion ◽  
Nonlinear Inverse Problem ◽  
Field Problem ◽  
Ill Posed

Our task is the identification of the reluctivity $\nu\,{=}\,\nu(B)$ in $\vec{H}\,{=}\,\nu(B) \vec{B}$, ($B\,{=}\,|\vec{B}|$) from measurements of the magnetic flux for different excitation currents in a driving coil, in the context of a nonuniform magnetic field distribution. This is a nonlinear inverse problem and ill-posed in the sense of unstable data dependence, whose solution is done numerically by a Newton type iterative scheme, regularized by an appropriate stopping criterion. The computational complexity of this method is determined by the number of necessary forward evaluations, i.e. the number of numerical solutions to the three-dimensional magnetic field problem. We keep the effort minimal by applying a special discretization strategy to the inverse problem, based on multigrid methods for ill-posed problems. Numerical results demonstrate the efficiency of the proposed method.

scholarly journals Numerical Solutions to Nonsmooth Dirichlet Problems Based on Lumped Mass Finite Element Discretization

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Haixiong Yu ◽  
Jinping Zeng
Keyword(s):  
Finite Element ◽  
Numerical Solutions ◽  
Piecewise Linear ◽  
Approximation Error ◽  
Uniform Mesh ◽  
Element Approximation ◽  
Dirichlet Problems ◽  
Lumped Mass ◽  
Element Discretization ◽  
Maximum Angle

We apply a lumped mass finite element to approximate Dirichlet problems for nonsmooth elliptic equations. It is proved that the lumped mass FEM approximation error in energy norm is the same as that of standard piecewise linear finite element approximation. Under the quasi-uniform mesh condition and the maximum angle condition, we show that the operator in the finite element problem is diagonally isotone and off-diagonally antitone. Therefore, some monotone convergent algorithms can be used. As an example, we prove that the nonsmooth Newton-like algorithm is convergent monotonically if Gauss-Seidel iteration is used to solve the Newton's equations iteratively. Some numerical experiments are presented.

scholarly journals Image reconstruction in optical tomography

1997 ◽  
Vol 352 (1354) ◽  
pp. 717-726 ◽  
Author(s):  
Simon R. Arridge ◽  
Martin Schweiger
Keyword(s):  
Inverse Problem ◽  
Numerical Solutions ◽  
Imaging Modality ◽  
Optical Tomography ◽  
Clinical Work ◽  
Forward Problem ◽  
Time Resolved ◽  
Reconstruction Methods ◽  
Solution Methods ◽  
Gradient Based

Optical tomography is a new medical imaging modality that is at the threshold of realization. A large amount of clinical work has shown the very real benefits that such a method could provide. At the same time a considerable effort has been put into theoretical studies of its probable success. At present there exist gaps between these two realms. In this paper we review some general approaches to inverse problems to set the context for optical tomography, defining both the terms forward problem and inverse problem . An essential requirement is to treat the problem in a nonlinear fashion, by using an iterative method. This in turn requires a convenient method of evaluating the forward problem, and its derivatives and variance. Photon transport models are described and methods for obtaining analytical and numerical solutions for the most commonly used ones are reviewed. The inverse problem is approached by classical gradient–based solution methods. In order to develop practical implementations of these methods, we discuss the important topic of photon measurement density functions , which represent the derivative of the forward problem. We show some results that represent the most complex and realistic simulations of optical tomography yet developed. We suggest, in particular, that both time–resolved, and intensity–modulated systems can reconstruct variations in both optical absorption and scattering, but that unmodulated, non–time–resolved systems are prone to severe artefact. We believe that optical tomography reconstruction methods can now be reliably applied to a wide variety of real clinical data. The expected resolution of the method is poor, meaning that it is unlikely that the type of high–resolution images seen in computed tomography or medical resonance imaging can ever be obtained. Nevertheless we strongly expect the functional nature of these images to have a high degree of clinical significance.

An inverse problem approach to determine possible memory length of fractional differential equations

2021 ◽  
Vol 24 (6) ◽  
pp. 1919-1936
Author(s):  
Chuan–Yun Gu ◽  
Guo–Cheng Wu ◽  
Babak Shiri
Keyword(s):  
Inverse Problem ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
Fundamental Problem ◽  
Real Life ◽  
Memory Length ◽  
Nonlinear Alternative ◽  
Starting Point ◽  
Fractional Differential

Abstract It is a fundamental problem to determine a starting point in fractional differential equations which reveals the memory length in real life modeling. This paper describes it by an inverse problem. Fixed point theorems such as Krasnoselskii’s and Schauder type’s and nonlinear alternative for single–valued mappings are presented. Through existence analysis of the inverse problem, the range of the initial value points and the memory length of fractional differential equations are obtained. Finally, three examples are demonstrated to support the theoretical results and numerical solutions are provided.

An inverse problem of determining the time-dependent potential in a higher-order Boussinesq-Love equation from boundary data

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Mousa Huntul ◽  
Mohammad Tamsir ◽  
Abdullah Ahmadini
Keyword(s):  
Inverse Problem ◽  
Boundary Data ◽  
Numerical Solutions ◽  
Physical Property ◽  
Neumann Boundary ◽  
Higher Order ◽  
Time Dependent ◽  
Potential Coefficient ◽  
Content Type ◽  
Dependent Potential

PurposeThe paper aims to numerically solve the inverse problem of determining the time-dependent potential coefficient along with the temperature in a higher-order Boussinesq-Love equation (BLE) with initial and Neumann boundary conditions supplemented by boundary data, for the first time.Design/methodology/approachFrom the literature, the authors already know that this inverse problem has a unique solution. However, the problem is still ill-posed by being unstable to noise in the input data. For the numerical realization, the authors apply the generalized finite difference method (GFDM) for solving the BLE along with the Tikhonov regularization to find stable and accurate numerical solutions. The regularized nonlinear minimization is performed using the MATLAB subroutine lsqnonlin. The stability analysis of solution of the BLE is proved using the von Neumann method.FindingsThe present numerical results demonstrate that obtained solutions are stable and accurate.Practical implicationsSince noisy data are inverted, the study models real situations in which practical measurements are inherently contaminated with noise.Originality/valueThe knowledge of this physical property coefficient is very important in various areas of human activity such as seismology, mineral exploration, biology, medicine, quality control of industrial products, etc. The originality lies in the insight gained by performing the numerical simulations of inversion to find the potential co-efficient on time in the BLE from noisy measurement.

Sign in / Sign up

Export Citation Format

Share Document